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Hamiltonian Circuits on 3-Polytopes JOURNAL OF COMBINATORIAL THEORY 9, 54-59 (1970) Hamiltonian Circuits on 3-Polytopes DAVID BARNETTE* AND ERNEST JUCOVI~ University of California, Davis, California 95616 and P. J. Safarik University, Kosice, Czechoslovakia Communicated by Victor Klee Received March 1968 ABSTRACT The smallest number of vertices, edges, or faces of any 3-polytope with no Hamiltonian circuit is determined. Similar results are found for simplicial polytopes with no Hamil- tonian circuit. In Grfinbaum [3] there appear the following questions by V. Klee: What is the minimum number of vertices, edges, and facets of a d-polytope or a simplicial d-polytope whose graph does not admit Hamiltonian circuits or paths? (A Hamiltonian path (circuit) of a graph G is a simple path (circuit) which involves all vertices of G. A d-polytope is called simplicial provided all its facets are (d -- 1)-simplices.) Theorems 3 and 4 of the present note contain an answer to this question for Hamiltonian circuits on 3-polytopes and simplicial 3-polytopes. First of all some necessary facts. Let G be a 3-connected graph embedded in the plane rr. By a face of G we mean the circuit of G bounding a connected component of ~" ,~ G. The graph G is said to be obtained from a graph G' by face splitting provided G is obtained from G' by adding an edge across a face of G'. Figure 1 shows the three ways in which this can be done. The inverse of face splitting is removing edges. An edge e of G is called removable provided the graph G ~-~ e is homeomorphic to some planar 3-connected graph G(e). We shall use the following two theorems of Steinitz [6]: THEOREM 1. A graph is 3-polyhedral (i.e., isomorphic to the graph of some 3-polytope) provided it is planar and 3-connected. * Research supported by NSF grant GP-8470. 54 HAMILTONIAN CIRCUITS ON 3-POLYTOPES 55 FIGURE 1 THEOREM 2. If G is a 3-polyhedral graph then there exists a sequence of graphs T = Gi , G2 .... , Gn-i , Gn = G such that Gi is obtained from Gi-i by face splitting, and T is the graph of the tetrahedron. An alternate formula- tion of Theorem 2 is that every 3-polyhedral graph G, G C= T has a remov- able edge. LEMMA 1. Let G be a 3-polyhedral graph, G ~& T. If F is a triangular face of G then F has a removable edge. PROOF: The proof is by induction on the number of edges of G. We begin the induction by examining the cases in which G is obtained from T by a face splitting. In this case G is either the graph of the pyramid over a quadrilateral or of the triangular prism, and by inspection we see that our result holds. If G is not obtained from T by a face splitting let e be a removable edge of G. If e is an edge of F we are done. If not, then F is a triangular face of G(e) and by induction F has an edge e' which is removable in G(e). Clearly e' is also removable in G. LEMMA 2. If G is a 3-polyhedral graph, G 5& T, and if F is a triangular face with a 3-valent vertex v, then the edge e off not meeting V is removable. PROOF: If G(e) is not 3-connected then two faces of G ~-~ e, say FI and F~, have a multiply connected union. One of these faces, say F1, must be the face created from the faces that contain e in G. The other face F~ must be a face of G and must meet F at the 3-valent vertex v. Any such face, however, meets both of the faces of G that contain e. This shows that two faces of G have a multiply connected union and thus G is not 3-connected which is a contradiction. LEMMA 3. If G is a 3-polyhedral graph, G :;& T, F is a triangular face of G, and G contains no Hamiltonian circuit then there is an edge e ofF such that G(e) has no Hamiltonian circuit. 56 BARNETTE AND JUCOVIC PROOF: Let e be a removable edge of F. If e does not meet a 3-valent vertex then each vertex of G is a vertex of G(e) and the lemma clearly holds. If e meets a 3-valent vertex v then the edge e' opposite v is removable and we are done, unless e' also meets a 3-valent vertex v'. In this ease we shall remove e. Now we see that, if a Hamiltonian circuit exists in G(e), one also exists in G. The different possibilities are illustrated in Figure 2, the dark lines are edges of a Hamiltonian circuit. G- Ce) C- i I G- eel G- G" (e) 6- FIOURE 2 THEOREM 3. The minimum number of vertices and edges or facets of a 3-polytope whose graph does not admit Hamiltonian circuits is 11 and 18 or 9, respectively. THEOREM 4. The minimum number of vertices and edges or facets of a simplicial 3-polytope whose graph does not admit Hamiltonian circuits is 11 and 27 or 18, respectively. PROOF: We show that if G is a 3-polyhedral graph which has fewer than 11 vertices then it has a Hamiltonian circuit. Because of Lemma 3 we may consider only those graphs without triangular faces. HAMILTONIAN CIRCUITS ON 3-POLYTOPES 57 CASE I. G has an n-valent vertex v, n >/5. In this case the set of faces meeting v contains at least 11 vertices. CASE II. G has a 4-valent vertex v. If one face meeting v has more than 4 edges then the set of faces meeting v has at least 10 vertices and it is easy to see that there must be at least one more vertex in G. If each face meeting v has 4 edges then the collection of faces meeting v has 9 vertices. If there is only one more vertex in G then G is the graph in Figure 3 which has a Hamiltonian circuit as indicated. FIGURE 3 CASE III. G is simple, i.e., G has only 3-valent vertices. The only simple 3-polyhedral graphs with 10 or fewer vertices and no triangular faces are the graph of the cube and the graph of the pentagonal prism, both of which have Hamiltonian circuits. We now show that if a 3-polyhedral graph G has fewer than 9 faces then it has a Hamiltonian circuit. Here also we consider only graphs without triangular faces. From Theorem 2, it follows that, if G has fewer than 8 faces, then it has fewer than 11 vertices, but we have shown that such a graph admits a Hamiltonian circuit. The same is true of octahedra which have more than one vertex of valency greater than 3. If G has 8 faces and 12 vertices then it is simple. Either it is the graph of the hexagonal prism or the graph which arises from that of a pentago- nal prism by splitting a pentagonal face. Both of these graphs have Hamiltonian circuits. The only 3-polyhedral graph with 8 faces and 11 vertices not having triangular faces is shown on Figure 4. It has a Hamiltonian circuit as indicated. It follows from Euler's formula (v -- e +f-~ 2) that if e < 18 then either v < 11 or f < 9. 58 BARNETTE AND JUCOVIC FIGURE 4 We now construct an example which attains these minimums. Let J' be the quadrivalent polytope with 11 facets obtained by "cutting" all 6 vertices of a triangular prism in such a way that the triangular facets in the places of original vertices have a common vertex (only one) if and only if the original vertices were neighbours. Thus no two new facets of J' in the places of original vertices have a common edge; neither do any two facets which correspond to the original facets have a common edge. The polytope J dual to J' (Figure 5) has 9 facets and 11 vertices which are divided into two classes T 1 , T2 in such a way that no two vertices of the same class are neighbours. Every circuit of the graph of J contains an even number of vertices and, therefore, J does not admit a Hamiltonian circuit. From Euler's formula follows the minimality of J also concerning the number of edges. So Theorem 3 is proved. Fiotm~ 5 By splitting all faces of the graph of the polytope J by edges connecting vertices of that class T1, which contains 5 vertices, we obtain a simplicial polytope J0 in which no two vertices from the class T~, which contains HAMILTONIAN CIRCUITS ON 3-POLYTOPES 59 6 vertices, are neighbors. Therefore, no circuit of the graph of the poly- tope J0 contains all vertices of T 2 and J0 does not have a Hamiltonian circuit, and it has the minimum number of vertices. As for a simplicial 3-polytope, f = 2v -- 4, e = ~-f, this polytope is minimal also concerning the number of edges and faces. (f, e, and v denote the number of facets, edges, and vertices of a 3-polytope, respectively.) So Theorem 4 is proved. REMARKS 1. At least a part of the assertion of Theorem 3 is contained without proof in Coxeter [2] and in Jucovi6 [4]. 2. The assertion of Theorem4is contained without proofinWhitney [9].
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